# Convex Sets

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Definition
 Convex Set A set $S$ is said to be convex if whenever two points $x$ and $y$ are in the set $S$ then all the points on the line segment joining $x$ and $y$ are also in the set $S$. A set $S$ is said to be convex if for $x$ and $y$ in $S$ and $\lambda$ is a real number such that $0\lt\lambda\lt1$, then $\lambda x+(1-\lambda)y$ always belongs to the set $S$. A convex set is always a closed set with bulging boundaries.